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The Non-explosive expansion material (NEEM) is a method more environmentally friendly than the harmful conventional rock fracturing techniques. However, it is slower and very costly. Thus, any means of economizing their use is very desirable. This paper investigates the crack growth between two neighboring holes of a gneiss rock internally pressurized by NEEM mixed with water with the aim to evaluate the influence of holes spacing (center-to-center distance), on the initiation and growth of cracks. Field experimental results reveal that crack starts earlier and grows faster with increasing ambient temperature. But when the ambient temperature is above 28 °C, the NEEM is “blown out” of the holes. At these ambient temperatures, the surrounding rocks are hot and cannot dissipate efficiently the heat generated by the hydration reaction. The best filling time was found to be in the evening when the daily hot temperature has drooped. The time to first crack increases as hole diameter decrease s . The 3D numerical modeling and simulation of crack growth between two neighboring holes internally pressurized by NEEM using ABAQUS (XFEM/CZM) software show s a good agreement with the theoretical and experimental results.

Rock fracturing process is one of the most important operations in quarry mining. Blasting is the most common method of rock splitting [

Nowadays, non-explosive rock fracturing methods are spreading in mining, especially in underground mining. Expansive cement, also known as Non-Explosive Expansive Material (NEEM), has been proved to be a safer, pollution free and silent rock splitting method [

When the stress intensity factor between two neighboring holes is equal to the rock fracture toughness, crack may be initiated [

As drawbacks, NEEM fracturing method is slower than blasting [

in the drilled holes, it takes a few hours (about 4 to 6 depending on the type of NEEM, the surrounding material, and ambient temperature) for the hydration peak to be reached [

In 2016, Natanzi and Leafer studied the influence of ambient temperature on the expansion pressure developed in a filled bore hole. They experimentally demonstrated that, higher temperature leads to greater and earlier expansive pressure as well as solid volumetric expansion of the NEEM. They failed to develop a mathematical model to explain the evolution of expansion pressure with ambient temperature [

In 2018, Shang et al. studied the influence of stress created by neighboring holes loaded with NEEM on the fracturing process. They proposed a mathematical model and determined the optimum hole diameter and spacing, but no experimental study or numerical simulation were carried out [

Holes spacing is one of the important parameters related to rock fracturing with NEEM. This is because optimum spacing results in improving the fragmentation process and reducing the cost [_{t}), the rock fracture toughness (K_{1c}) of mode 1, the hole diameter (d), and the expansion pressure (P) as shown in Equation (1) [

S = { − 0.0888 ( P σ t ) 2 + 1.0824 ( P σ t ) − 2.1583 } P 2 d 2 K I c 2 (1)

Even though this model considers more parameters to calculate the holes spacing, it fails to consider the influence of the confining stress acting on the rock mass and also the evolution of expansion pressure (P) with loading time as predicted by many studies. According to Gholinejad et al. in 2012, the expansion pressure (P) is related to three parameters: time, holes diameter and the rock fracturing toughness of mode I as in Equation (2) [

P = 0.12 r 0 0.407 t 0.933 ( 37 K I c − 11.6 ) 0.493 (2)

In recent years, many numerical simulations have focused on the investigation of stress, crack initiation and propagation around NEEM loaded holes [

Numerically, using cohesive zone model for brittle material with an assumption of some plasticity is found to be a good approach to predict the crack growth in rocks [

In 2014, Zolenka et al. used combined pressure deformation cohesive zone Method (CZM) and Extended Finite Element Method (XFEM) to model and simulate cracks and fracture propagation during hydraulic fracturing of rocks [

This paper combines the XFEM approach with cohesive zone model (CZM) to analyze the crack growth between two neighboring holes loaded with NEEM of a Gneiss hard brittle rock using ABAQUS software with the objective of determining the optimum spacing in order to minimize the rock fracturing project cost. The numerical results are compared with those achieved experimentally and with theoretical results available in the literature.

Experimental field works were carried out in a Gneiss quarry at “Nkolondom” in the Centre region of Cameroon during the colder season when the ambient temperature varies from 16˚C to 24˚C during the day and 14˚C to 19˚C during the night. During this period, the ambient temperature varies very little during the three sections of the day: the morning (from 5.30 am to 10.30 am); in the afternoon (from 12:30 p.m. to 3 p.m.); and in the evening (6.30 p.m. to 9 p.m.). Thus, we assumed a constant temperature during each of the slices of the day by considering the average temperature. Natanzi et al. (2016), concluded that hydration peak is reached after 4 to 6 hours after filling the predrilled holes with NEEM. We will therefore fill the holes at the launch of a day section as sliced previously, this will allow us to reach the hydration peak being in the same slice, before the ambient temperature varies considerably.

Gneiss is the predominant metamorphic rock of this area and is usually used as building stones and as ornamental stones for floor and wall tiling. Holes were drilled on an outcrop and rock blocks with a pneumatic rock drill with three drill bits: 30 mm, 40 mm and 50 mm. Measuring tools (electronic caliper, electronic meter, chronometer…) were used for the measurement of crack lengths and widths.

In order to study the effects of anisotropy, drilled holes were spaced perpendicular and parallel to the foliation planes (weakness planes) of the gneiss rock.

The NEEM used during these experiments is CRACKAG that is manufactured in China. Expansive cement usually has a chemical composition as shown in

NEEM is delivered from the manufacturer in a powder form and mixed with water to form slurry. The slurry is poured into drilled holes in rock. Water was poured in the powder expansive grout (in the adequate proportion) and mixed with a chemical electric liquid mixer. The mixing time was about five minutes to achieve a good homogeneity of the slurry. It was then poured in the drilledholes.

Chemical component | Percentage by Mass (%) |
---|---|

CaO | 81 - 96 |

SiO_{2} | 1.5 - 8.5 |

Al_{2}O_{3} | 0.3 - 5.5 |

Fe_{2}O_{3} | 0.2 - 3 |

MgO_{2} | 0 - 1.6 |

SO_{3} | 0.6 - 4.6 |

Considering a volume element of the borehole located at a distance of r from the center of the first internally pressurized hole. The stress distribution (due to the internal pressure) is as shown in _{r} is the radial stress, σ_{θ} the orthoradial stress, and τ_{rθ} the shear stress. The circular holes, internally pressurized by NEEM are of the same radius r_{0} with the spacing between the two holes of S.

Based on the Theory of elasticity [

( σ r + ∂ σ r ∂ r d r ) ( r + d r ) d θ − σ r r d θ − ( σ θ + ∂ σ θ ∂ θ d θ ) d r sin ( d θ 2 ) − σ θ d r sin ( d θ 2 ) + ( τ r θ + ∂ τ r θ ∂ θ d θ ) d r cos ( d θ 2 ) − τ r θ d r cos ( d θ 2 ) = 0 (3)

From Equation (3), we can obtain the following differential equations

{ ∂ σ r ∂ r + 1 r ∂ τ r θ ∂ θ + σ r − σ θ r = 0 1 r ∂ σ θ ∂ θ + ∂ τ r θ ∂ r + 2 τ r θ r = 0 (4)

When the borehole is internally pressurized, the internal pressure is uniform on the hole walls, thus the radial deformation is uniform ( τ r θ = 0 ), and Equation (6) becomes

{ ∂ σ r ∂ r + σ r − σ θ r = 0 ∂ σ θ ∂ θ = 0 (5)

The solutions of Equation (5) is in the form of

σ r = C r n , (6)

where C and n are constants.

The boundary conditions are as follows:

{ σ r = P when r = r 0 σ r = 0 when r = S (7)

Equations (5) and (6) then give the following solutions:

σ r = r 0 2 P ( 1 − S 2 r ) S 2 − r 0 2 , σ θ = r 0 2 P ( 1 + S 2 r ) S 2 − r 0 2 (8)

With P the internal pressure generated by the NEEM, S the spacing between the two holes, r_{0} the radius of the holes.

Substituting Equation (2) into Equation (8), we obtain:

σ r = r 0 2 ( 0.12 r 0 0.407 t 0.933 ( 37 K I c − 11.6 ) 0.493 ) ( 1 − S 2 r ) S 2 − r 0 2 ; σ θ = r 0 2 ( 0.12 r 0 0.407 t 0.933 ( 37 K I c − 11.6 ) 0.493 ) ( 1 + S 2 r ) S 2 − r 0 2 (9)

According to

σ y y = σ r sin 2 θ + σ θ cos 2 θ (10)

The viewpoint from which cohesive zone models originate regards fracture as a gradual phenomenon in which separation takes place across an extended crack tip, or cohesive zone, and is resisted by cohesive tractions [

The idea of CZM is based on the assumption that the material’s failure process during fracture is limited to a narrow band in front of the main crack (Kuna et al.

2013). In CZM, the material follows the traction-separation law used for defining the shear traction and crack sliding displacement relations across the crack tip. Before the first principal stress reaches the tensile strength, the material behaves linearly elastic. As soon as the tensile strength is exceeded, the material begins to fail and the crack will get initiated. Crack initiation refers to the beginning of degradation of the cohesive response at an enriched element. The process of degradation begins when the stresses or the strains satisfy specified crack initiation criteria [

When crack grows, the cohesive zone elements assigned in the mesh opens to simulate crack initiation. Since the crack path only follows the cohesive zone elements, crack propagation strongly depends on the mesh of the cohesive zone elements. This leads to the inclusion of Extended Finite Element Method (XFEM) approach, where the crack geometry is overlapped over the crack domain and their propagation happens without depending on the mesh.

The extended finite element method (XFEM) is a numerical technique which extends the classical finite element method approach focusing on crack propagation problems. The main idea behind this method is to deal with simple meshes and to take into account discontinuous displacements inside a finite element. Extended finite element methods (XFEM) allows simulation of crack growth without re meshing [

Let’s consider 𝑥, a point in a finite element that is intersected by a crack. To calculate the displacement at point 𝑥 located within the domain, the approximation for a displacement vector function is,

U ( x ) = ∑ i ∈ N N i ( x ) u i ︸ Classical FEM approximation + ∑ i ∈ N d N i ( x ) H ( x ) a i ︸ Discontinuous enrichment + ∑ i ∈ N p N i ( x ) ( ∑ j = 1 4 F j ( x ) b i j ) ︸ Crack tip enrichment (11)

where N is the total nodes of the mesh,

N d ∈ N are all the enriched nodes by the discontinuity,

N p ∈ N are all the nodes near the crack tip,

N_{i}(x) are the usual nodal shape functions,

u_{i} are the displacement degrees of freedom at node i,

a_{i} are the nodal enriched degree of freedom vector,

H(x) is the Heaviside function associated to the discontinuous jump,

b i j are the nodal enriched degree of freedom vector and

F_{j}(x) are the elastic asymptotic crack-tip functions and are given by the following expression:

( F j ( x ) ) = ( r sin ( θ 2 ) ; r cos ( θ 2 ) ; r sin ( θ 2 ) sin ( θ ) ; r cos ( θ 2 ) sin ( θ ) ) (12)

(r, θ) are the polar co-ordinates related to the local axis of the crack tip and can be expressed in terms of the level sets as follows:

r = l s n 2 + l s t 2 , θ = arctan ( l s t l s n ) (13)

These functions form the basis of the asymptotic field 1/r around the crack tip, and introduce additional degrees of freedom in each node, improving the solution accuracy near the crack tip. The first function r sin ( θ / 2 ) is discontinuous along the crack surfaces, giving the effect of required discontinuity in the approximation along the crack, as seen in

With the use of the above mentioned near-tip enrichment functions an element partially cut by the crack can be modeled (Ahmed A., 2009).

Full XFEM enrichment is used only for the simulation of stationary cracks. The Near-tip asymptotic singularity is not considered for crack growth numerical analysis. Thus, only the displacement jump across a cracked element is considered in this study.

Equation (1) which is useful to determine the holes spacing is very complex and nonlinear. This is because the expansive pressure is one of the parameters of the equation, and it is well known that this pressure is dependent on parameters such as time, rock fracture toughness and holes radius as revealed by Equation (2), but also by ambient temperature [

The Gneiss rock parameters used to describe CZM in ABAQUS are: Young’s modulus E, the Poisson’s ratio ν, the shear modulus G, the rock fracture toughness of mode 1, the tensile strength σ_{t} and the rock density.

In

Damage initiation criterion used for this study is the Maximum principal stress criterion (MAXPS).Until the crack gets initiated, the material adopts the elastic properties. Once the material strength reaches its material limit, it will behave based on traction-separation law. Crack initiation occurs when the maximum

Gneiss Rock parameters | Value |
---|---|

Density | 2.7 |

Young’s modulus E (GPa) | E_{1} = 12.5; E_{2} = 12; E_{3} = 12.5 |

Poisson’s ratio ν | ν_{1} = 0.35; ν_{2} = 0.24; ν_{3} = 0.3 |

Shear modulus G (GPa) | G_{1} = 4.6; G_{2} = 4.75; G_{3} = 4.86 |

Tensile strength is σ_{t} (MPa) | 10.9 |

Mode 1 fracture toughness (MPa.mm^{1/2}) | 0.3 |

principal stress reaches critical value. In regards to various calibrations that have been noted in different literatures (Chen, 2013) [_{0} (yield stress (σ_{𝑦})) and the Cohesive Energy Γ_{0} (fracture energy (J_{𝐼𝐶})).

The drilled hole was internally pressurized conferring to Equation (2). Crack growth simulation was carried out on the outcrop. Thus, the initial conditions are: three degree of freedom as shown in

1) Influence of ambient temperature

Drilled Holes on the outcrop of diameter 50 mm were filled during three different hours of the day: In the morning with an ambient temperature of 20˚C, at midday with an ambient temperature of 28˚C, at the evening with an ambient temperature of 22˚C. The mixture temperature was 7˚C (NEEM was mixed with cold water). The experiments were carried out several times and the following results were achieved. Although the manufacturer recommended that the expansive cement could be used at temperatures of 25˚C to 40˚C, the NEEM was “blowout” of the holes filled at midday (ambient temperature of 28˚C).

the time for the first crack to appear was 8 hours for the morning filled holes and 10 hours 30 minutes for the evening filled holes.

From this figure, it appears that crack grows faster with increasing ambient temperature. Thus, the best filling time is in the evening when the daily hot temperature has dropped. Though the cracks started latter than of the holes filled in the morning, the propagations were faster, this may be because usually the ambient temperature drops in the evening, and night are colder.

The best time to first crack was achieved for holes filled at ambient temperature of 20˚C and at 22˚C, the “blown out» phenomenon did not happen as predicted by Natanzi et al. [

CaO + H 2 O → Ca ( OH ) 2 + 15.2 ( kcal / mol ) ↑ (14)

Nonetheless, the “blown out” phenomenon occurred after 4 hours for holes filled during midday when the ambient temperature was 28˚C, this may be because the surrounding rocks were hot (due to sun heating from morning to midday), and could not dissipate efficiently the heat during hydration reaction [

2) Influence of holes diameters

Holes of diameter 30 mm, 40 mm and 50 mm were drilled and filled on the outcrop.

From

3) Influence of rock Anisotropy.

Gneiss is an anisotropic hard brittle rock. Holes of diameters 40 mm and 50 mm were drilled on the two blocks along the foliation lines.

crack growth with time for each borehole diameter.

4) Numerical results

Aiming for a comparative analysis between the experimental results and the simulation, holes of diameters 30, 40 and 50 mm were drilled on the cohesive zone. Spacing S used during the study varies for each drilled hole from 120 to 250 mm. _{a} and t_{b} in hours respectively, with t_{a} < t_{b}. The red color in the figure means high stress concentration.

Figures 16-18 display the comparative study between experimental and numerical simulation results. They reveal that the numerical solution converges with experimental field results. These figures also illustrate that cracks initiate very much earlier numerically than experimentally (time to first crack for a hole of diameter 40 mm is 10 hours experimentally and 1 hour 30 min for numerical simulation). This is because ABAQUS software does not consider the hydration time of the expansive cement. In fact, when the NEEM is mixed with water and poured in the drilled holes, it takes few hours (about 4 to 6 depending on the type of NEEM powder, surrounding material and the ambient temperature) for

the hydration peak to be reached [

The hydration process depends on the type of NEEM powder (percentage of CaO) and ambient temperature (Natanzi et al. 2016). However, a quantitative relation/equation between ambient temperature, NEEM type and the performance of NEEM in terms of expansive pressure, is still not available to our knowledge. Such further work needs to be performed in this regard to further improve the prediction performance of Equation (2) (Shang et al. in 2018), and reproduce the nature of the expansive grout, that takes several hours for the internal pressure to become considerable.

Though the in-situ pressures were neglected, rock modeling and crack numerical simulation with XFEM method exhibit some similarities with the experimental field results.

5) Optimum hole spacing determination

From the simulation of crack growth as shown in

Figures 20-22 illustrate the evolution of the tensile stress at the midpoint for holes of diameter 30 mm, 40 mm and 50 mm respectively. The time after NEEM loading considered is: 4, 8, 12, 16, 20, and 25 h.

When the maximum principal stress reaches critical value (greater than the tensile strength of the rock), crack initiation occurs and propagate. Figures 20-22,

thus give us the optimum spacing for a particular diameter, and also reveal the corresponding fragmentation time. For example,

From Figures 20-22, optimum holes spacing evolution with fragmentation time and radius can be deduced as displayed in

From _{r} that corresponds to a particular radius and the desired fragmentation time as given by Equation (15).

{ S 15 = − 0.0833 t 2 + 4.25 t + 75.833 S 20 = − 0.001 t 2 + 1.0068 t + 152.22 S 25 = − 0.0017 t 2 + 1.2173 t + 195.61 (15)

Hole spacing is an important parameter which influences the rock fragmentation process, in fact when the holes are too close (small hole spacing), cracks will occur and grow rapidly but the project cost will be very high because more holes will need to be drilled. In the other hand, large spacing will result in few holes to drill, but crack growth will be delayed or may not even happen. Optimal spacing as given by Equating (15) therefore helps to predict the fragmentation time for a radius at the adequate spacing. This study focused only on three diameters, but more simulation with XFEM can be done to obtain S_{r} relation with fragmentation time for other desired radii.

Figures 20-22 are similar to those obtained at the same position ( r = S / 2 ; θ = 0 ) with analytical method by Shang et al. in 2018. The analytical equation used by Shang et al. to determine the maximum principal stress is very complex and is nonlinear when the azimuth angle, θ is not equal to zero. With this coupled simulation method (CZM and XFEM) using ABAQUS software, it is therefore possible to determine the principal stress at any point around the internally pressurized holes with the desired azimuth.

In this paper, experimental field works and numerical simulation were carried out to investigate the crack growth between two neighboring holes of a gneiss rock internally pressurized by NEEM. Experimental results reveal that ambient temperature, hole diameter, rock anisotropy and holes spacing influence the crack growth. It appears that crack grows faster with increasing temperature, but when the ambient temperature is above 22˚C, NEEM will be blown out of the holes. The numerical models simulated with ABAQUS software (XFEM coupled with CZM) of crack growth between two neighboring holes internally pressurized by NEEM show a good agreement with the theoretical/analytical and experimental results.

Further study will be carried out to investigate on the heat dissipation on surrounding rocks during the hydration reaction.

The data used to support the findings of this study are available from the corresponding author upon request.

The initial draft of the article was done by Frank Ferry KAMGUE TIAM. The drafted manuscript was reviewed and corrected by Noël KONAÏ and Lucien MEVA’A. The concept was conceived by Raïdandi DANWE and he also carried out the final check of the paper.

The authors declare no conflicts of interest regarding the publication of this paper.

Tiam, F.F.K., Danwe, R., Konaï, N. and Meva’a, L. (2020) Experimental Study and Numerical Simulation Using Extended Finite Element Method (XFEM) Combined with Cohesive Zone Model (CZM), of Crack Growth Induced by Non-Explosive Expansive Material on Two Neighboring Circular Holes of A Gneiss Rock. Open Journal of Applied Sciences, 10, 592-612. https://doi.org/10.4236/ojapps.2020.1010042